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  <div class="section" id="method-and-theory">
<h1>Method and Theory<a class="headerlink" href="#method-and-theory" title="Permalink to this headline">¶</a></h1>
<div class="section" id="stroh-theory">
<h2>Stroh theory<a class="headerlink" href="#stroh-theory" title="Permalink to this headline">¶</a></h2>
<p>A detailed description of the Stroh method to solve the Eshelby solution
for an anisotropic straight dislocation can be found in the <a class="reference external" href="https://www.ctcms.nist.gov/potentials/atomman/">atomman
documentation</a>.</p>
</div>
<div class="section" id="orientation">
<h2>Orientation<a class="headerlink" href="#orientation" title="Permalink to this headline">¶</a></h2>
<p>One of the most important considerations in defining an atomistic system
containing a dislocation monopole system is the system’s orientation. In
particular, care is needed to properly align the system’s Cartesian
axes, <span class="math notranslate nohighlight">\(x, y, z\)</span>, the system’s box vectors, <span class="math notranslate nohighlight">\(a, b, c\)</span>, and
the Stroh solution’s axes, <span class="math notranslate nohighlight">\(u, m, n\)</span>.</p>
<ul class="simple">
<li><p>In order for the dislocation to remain perfectly straight in the
atomic system, the dislocation line, <span class="math notranslate nohighlight">\(u\)</span>, must correspond to
one of the system’s box vectors. The resulting dislocation monopole
system will be periodic along the box direction corresponding to
<span class="math notranslate nohighlight">\(u\)</span>, and non-periodic in the other two box directions.</p></li>
<li><p>The Stroh solution is invariant along the dislocation line direction,
<span class="math notranslate nohighlight">\(u\)</span>, thereby the solution is 2 dimensional. <span class="math notranslate nohighlight">\(m\)</span> and
<span class="math notranslate nohighlight">\(n\)</span> are the unit vectors for the 2D axis system used by the
Stroh solution. As such, <span class="math notranslate nohighlight">\(u, m\)</span> and <span class="math notranslate nohighlight">\(n\)</span> are all normal to
each other. The plane defined by the <span class="math notranslate nohighlight">\(um\)</span> vectors is the
dislocation’s slip plane, i.e. <span class="math notranslate nohighlight">\(n\)</span> is normal to the slip plane.</p></li>
<li><p>For LAMMPS simulations, the system’s box vectors are limited such
that <span class="math notranslate nohighlight">\(a\)</span> is parallel to the <span class="math notranslate nohighlight">\(x\)</span>-axis, and <span class="math notranslate nohighlight">\(b\)</span> is in
the <span class="math notranslate nohighlight">\(xy\)</span>-plane (i.e. has no <span class="math notranslate nohighlight">\(z\)</span> component). Based on this
and the previous two points, the most convenient, and therefore the
default, orientation for a generic dislocation is to</p>
<ul>
<li><p>Make <span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(a\)</span> parallel, which also places <span class="math notranslate nohighlight">\(u\)</span>
parallel to the <span class="math notranslate nohighlight">\(x\)</span>-axis.</p></li>
<li><p>Select <span class="math notranslate nohighlight">\(b\)</span> such that it is within the slip plane. As
<span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(a\)</span> must also be in the slip plane, the plane
itself is defined by <span class="math notranslate nohighlight">\(a \times b\)</span>.</p></li>
<li><p>Given that neither <span class="math notranslate nohighlight">\(a\)</span> nor <span class="math notranslate nohighlight">\(b\)</span> have <span class="math notranslate nohighlight">\(z\)</span>
components, the normal to the slip plane will be perpendicular to
<span class="math notranslate nohighlight">\(z\)</span>. From this, it naturally follows that <span class="math notranslate nohighlight">\(m\)</span> can be
taken as parallel to the <span class="math notranslate nohighlight">\(y\)</span>-axis, and <span class="math notranslate nohighlight">\(n\)</span> parallel to
the <span class="math notranslate nohighlight">\(z\)</span>-axis.</p></li>
</ul>
</li>
</ul>
</div>
<div class="section" id="calculation-methodology">
<h2>Calculation methodology<a class="headerlink" href="#calculation-methodology" title="Permalink to this headline">¶</a></h2>
<ol class="arabic simple">
<li><p>An initial system is generated based on the loaded system and <em>uvw</em>,
<em>atomshift</em>, and <em>sizemults</em> input parameters. This initial system is
saved as base.dump.</p></li>
<li><p>The atomman.defect.Stroh class is used to obtain the dislocation
solution based on the defined <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(n\)</span> vectors,
<span class="math notranslate nohighlight">\(C_{ij}\)</span>, and the dislocation’s Burgers vector, <span class="math notranslate nohighlight">\(b_i\)</span>.</p></li>
<li><p>The dislocation is inserted into the system by displacing all atoms
according to the Stroh solution’s displacements. The dislocation line
is placed parallel to the specified <em>dislocation_lineboxvector</em> and
includes the Cartesian point (0, 0, 0).</p></li>
<li><p>The box dimension parallel to the dislocation line is kept periodic,
and the other two box directions are made non-periodic. A boundary
region is defined that is at least <em>bwidth</em> thick at the edges of the
two non-periodic directions, such that the shape of the active region
corresponds to the <em>bshape</em> parameter. Atoms in the boundary region
are identified by altering their integer atomic types.</p></li>
<li><p>The dislocation is relaxed by performing a LAMMPS simulation. The
atoms in the active region are allowed to relax either with nvt
integration followed by an energy/force minimization, or with just an
energy/force minimization. The atoms in the boundary region are held
fixed at the elastic solution. The resulting relaxed system is saved
as disl.dump.</p></li>
<li><p>Parameters associated with the Stroh solution are saved to the
results record.</p></li>
</ol>
</div>
</div>


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  <h3><a href="../../index.html">Table of Contents</a></h3>
  <ul>
<li><a class="reference internal" href="#">Method and Theory</a><ul>
<li><a class="reference internal" href="#stroh-theory">Stroh theory</a></li>
<li><a class="reference internal" href="#orientation">Orientation</a></li>
<li><a class="reference internal" href="#calculation-methodology">Calculation methodology</a></li>
</ul>
</li>
</ul>

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